$f$ differentiable and $f(0)=f(1)=0$. , prove that $|f'(x)| \le \frac{A}{2}$ $\forall x \in [0,1]$

Question

Let $f$ be differentiable on $[0,1]$ and $f(0)=f(1)=0$.

Also, we know $|f''(x)| \le A$ on $(0,1)$, prove that $|f'(x)| \le \frac{A}{2}$ $\forall x \in [0,1]$

I'm guessing I should use taylor expansion but have no idea how to start

Any hints/suggestions? thanks

Answer 1

You don't need Taylor. Rolle's Theorem guarantees a $c\in (0,1)$ s.t. $f'(c)=0$. Now use Fundamental Theorem of Calculus for...